The PI's research is in classical convexity theory and convex geometric analysis. A primary goal is to get a better understanding of the structure of convex bodies. To do so she uses techniques from different areas of mathematics: analysis, differential geometry, convexity theory, probability theory. She investigates isoperimetric inequalities and affine isoperimetric inequalities. These provide powerful tools in characterizing and classifying convex sets. Through her investigation of the affine surface area -originally a concept of affine differential geometry and occurring in the affine isoperimetric inequality- she was lately led to an extensive study of questions of approximation of convex bodies by polytopes . The affine surface area appears naturally in this context as it is related to the boundary structure of a convex body. The PI has investigated and still is investigating different aspects of approximation of convex bodies by polytopes. In one paper, for instance, she -together with her collaborator- proved the surprising result that random approximation by polytopes (choosing the vertices of the approximating polytope randomly on the boundary of the body) is as good as best approximation. Besides convexity tools, probabilistic tools, like concentration of measure, have proved to be very efficient in convexity. The PI continues her investigation of such probabilistic results for advancing her research in structural results in convexity and its applications to local Banach space theory and quantum information theory.
Past experience has led the PI to believe that purely theoretical concepts are also useful in applications. She has experienced that the methods and results from these areas find applications in other fields of mathematics and in applied areas: Geometric tomography, a tool having its origins in classical convexity theory, gives a method to recover convex shapes from its sections or projections. This is used in computer vision and image analysis, in biology and medicine where convex shapes (organs) occur naturally. Geometric algorithms find applications in computer science. To be more explicit, a mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, if you want to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow us to predict its future motion from position and velocity we would need six separate numbers in all. If you want to track 100 distinct molecules of the air in the room then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases, and this grant will study some of these patterns that are recent discoveries.
